561 research outputs found

    The triangulated categories of framed bispectra and framed motives

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    An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory SH(k)SH(k) is suggested. The triangulated category of framed bispectra SHnisfr(k)SH_{nis}^{fr}(k) and effective framed bispectra SHnisfr,eff(k)SH_{nis}^{fr,eff}(k) are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that SHnisfr(k)SH_{nis}^{fr}(k) and SHnisfr,eff(k)SH_{nis}^{fr,eff}(k) recover the classical Morel-Voevodsky triangulated categories of bispectra SH(k)SH(k) and effective bispectra SHeff(k)SH^{eff}(k) respectively. We also recover SH(k)SH(k) and SHeff(k)SH^{eff}(k) as the triangulated category of framed motivic spectral functors SHS1fr[Fr0(k)]SH_{S^1}^{fr}[\mathcal Fr_0(k)] and the triangulated category of framed motives SHfr(k)\mathcal {SH}^{fr}(k) respectively constructed in the paper

    Reconstructing projective schemes from Serre subcategories

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    Given a positively graded commutative coherent ring A which is finitely generated as an A_0-algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y\subseteq Proj A of the form Y=\bigcup_{i\in\Omega}Y_i with quasi-compact open complement Proj A\Y_i for all i\in\Omega is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces (Proj A,O_{Proj A}) --> (Spec(qgr A),O_{qgr A}), where (Spec(qgr A),O_{qgr A}) is a ringed space associated to the lattice L_{serre}(qgr A) of tensor Serre subcategories of qgr A.Comment: some minor corrections mad

    Torsion classes of finite type and spectra

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    Given a commutative ring R (respectively a positively graded commutative ring A=\ps_{j\geq 0}A_j which is finitely generated as an A_0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y\subset Spec R (respectively Y\subset Proj A) of the form Y=\cup_{i\in\Omega}Y_i, with Spec R\Y_i (respectively Proj A\Y_i) quasi-compact and open for all i\in\Omega, is established. Using these bijections, there are constructed isomorphisms of ringed spaces (Spec R,O_R)-->(Spec(Mod R),O_{Mod R}) and (Proj A,O_{Proj A})-->(Spec(QGr A),O_{QGr A}), where (Spec(Mod R),O_{Mod R}) and (Spec(QGr A),O_{QGr A}) are ringed spaces associated to the lattices L_{tor}(Mod R) and L_{tor}(QGr A) of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes perf(R) and the torsion classes of finite type in Mod R is established
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